The ability to precisely and coherently control atomic systems has improved dramatically in the last two decades, driving remarkable advancements in quantum computation and simulation. In recent years, atomic and atom-like systems have also been served as a platform to study topological phases of matter and non-equilibrium many-body physics. Integrated with rapid theoretical progress, the employment of these systems is expanding the realm of our understanding on a range of physical phenomena. In this dissertation, I draw on state-of-the-art experimental technology to develop several new ideas for controlling and applying atomic systems. In the first part of this dissertation, we propose several novel schemes to realize, detect, and probe topological phases in atomic and atom-like systems. We first theoretically study the intriguing properties of Hopf insulators, a peculiar type of topological insulators beyond the standard classification paradigm of topological phases. Using a solid-state quantum simulator, we report the first experimental observation of Hopf insulators. We demonstrate the Hopf fibration with fascinating topological links in the experiment, showing clear signals of topological phase transitions for the underlying Hamiltonian. Next, we propose a feasible experimental scheme to realize the chiral topological insulator in three dimensions. They are a type of topological insulators protected by the chiral symmetry and have thus far remained unobserved in experiment. We then introduce a method to directly measure topological invariants in cold-atom experiments. This detection scheme is general and applicable to probe of different topological insulators in any spatial dimension. In another study, we theoretically discover a new type of topological gapless rings, dubbed a Weyl exceptional ring, in three-dimensional dissipative cold atomic systems. In the second part of this dissertation, we focus on the application of atomic systems in quantum computation and simulation. Trapped atomic ions are one of the leading platforms to build a scalable, universal quantum computer. The common one-dimensional setup, however, greatly limits the system's scalability. By solving the critical problem of micromotion, we propose a two-dimensional architecture for scalable trapped-ion quantum computation. Hamiltonian tomography for many-body quantum systems is essential for benchmarking quantum computation and simulation. By employing dynamical decoupling, we propose a scalable scheme for full Hamiltonian tomography. The required number of measurements increases only polynomially with the system size, in contrast to an exponential scaling in common methods. Finally, we work toward the goal of demonstrating quantum supremacy. A number of sampling tasks, such as the boson sampling problem, have been proposed to be classically intractable under mild assumptions. An intermediate quantum computer can efficiently solve the sampling problem, but the correct operation of the device is not known to be classically verifiable. Toward practical verification, we present an experimental friendly scheme to extract useful and robust information from the quantum boson samplers based on coarse-grained measurements. In a separate study, we introduce a new model built from translation-invariant Ising-interacting spins. This model possesses several advantageous properties, catalyzing the ultimate experimental demonstration of quantum supremacy.
The ability to precisely and coherently control atomic systems has improved dramatically in the last two decades, driving remarkable advancements in quantum computation and simulation. In recent years, atomic and atom-like systems have also been served as a platform to study topological phases of matter and non-equilibrium many-body physics. Integrated with rapid theoretical progress, the employment of these systems is expanding the realm of our understanding on a range of physical phenomena. In this dissertation, I draw on state-of-the-art experimental technology to develop several new ideas for controlling and applying atomic systems. In the first part of this dissertation, we propose several novel schemes to realize, detect, and probe topological phases in atomic and atom-like systems. We first theoretically study the intriguing properties of Hopf insulators, a peculiar type of topological insulators beyond the standard classification paradigm of topological phases. Using a solid-state quantum simulator, we report the first experimental observation of Hopf insulators. We demonstrate the Hopf fibration with fascinating topological links in the experiment, showing clear signals of topological phase transitions for the underlying Hamiltonian. Next, we propose a feasible experimental scheme to realize the chiral topological insulator in three dimensions. They are a type of topological insulators protected by the chiral symmetry and have thus far remained unobserved in experiment. We then introduce a method to directly measure topological invariants in cold-atom experiments. This detection scheme is general and applicable to probe of different topological insulators in any spatial dimension. In another study, we theoretically discover a new type of topological gapless rings, dubbed a Weyl exceptional ring, in three-dimensional dissipative cold atomic systems. In the second part of this dissertation, we focus on the application of atomic systems in quantum computation and simulation. Trapped atomic ions are one of the leading platforms to build a scalable, universal quantum computer. The common one-dimensional setup, however, greatly limits the system's scalability. By solving the critical problem of micromotion, we propose a two-dimensional architecture for scalable trapped-ion quantum computation. Hamiltonian tomography for many-body quantum systems is essential for benchmarking quantum computation and simulation. By employing dynamical decoupling, we propose a scalable scheme for full Hamiltonian tomography. The required number of measurements increases only polynomially with the system size, in contrast to an exponential scaling in common methods. Finally, we work toward the goal of demonstrating quantum supremacy. A number of sampling tasks, such as the boson sampling problem, have been proposed to be classically intractable under mild assumptions. An intermediate quantum computer can efficiently solve the sampling problem, but the correct operation of the device is not known to be classically verifiable. Toward practical verification, we present an experimental friendly scheme to extract useful and robust information from the quantum boson samplers based on coarse-grained measurements. In a separate study, we introduce a new model built from translation-invariant Ising-interacting spins. This model possesses several advantageous properties, catalyzing the ultimate experimental demonstration of quantum supremacy.
(English) In recent years, the evolution of quantum technologies has resulted in an unprecedented control over individual quantum particles and many-body systems. This remarkable progress has given rise to a new era, marked by the convergence of classical and quantum methodologies to investigate fundamental questions concerning the nature of quantum matter, improving our understanding of the role of entanglement in solid-state systems or the mechanisms behind high-energy physics. From analog quantum simulators to digital quantum computers, these advancements hold potential across diverse domains. This thesis explores the interplay between classical and quantum methods in understanding topological phases of matter. We concentrate on three distinct directions: non-conventional topological superconductors, interaction-induced topological phases in ultracold atom quantum simulators, and applications of variational quantum algorithms. Each trajectory relies on the combination of different techniques with the aim of understanding and characterizing topological phenomena in different settings. Exploring non-conventional topological superconductors involves extending the paradigmatic Kitaev chain model by incorporating additional terms in the Hamiltonian such as long-range interactions and quasi-periodic potentials. This investigation is relevant to better understand the impact of real-world conditions on the both the topological and localization properties of systems hosting non-local Majorana modes, which are promising candidates for topological quantum computation. In the realm of interacting systems, we explore the realization of interaction-induced topological phases in systems of ultracold atoms in optical lattices, both in one and two dimensions. The remarkable control and versatility of such platforms enable the simulation of both theoretical topological models and strongly correlated physics. Notably, the interplay between interactions and topology can give rise to intriguing phenomena, such as delocalized fractional charges and gapless topological phases,challenging existing intuition. We employ advanced numerical methods based on tensor networks to benchmark the experimental proposals that open the door to the realization and detection of novel many-body phases of matter, including topological quantum critical points and a higher-order topological Peierls insulator in Bose-Hubbard models with long-range interactions. Variational quantum algorithms, conversely, have the potential to efficiently tackle a wide range of problems, including ground state search, phase classification or accessing topological invariants. Despite current limitations in trainability and scalability, these hybrid classical-quantum algorithms provide practical insights into current quantum hardware capabilities and can inspire future architectures. We explore the application of variational quantum algorithms to shed light on topological phenomena, raising questions about their ability to discern topological phase transitions and compute topological invariants in situations where classical approaches fail. This thesis presents a comprehensive exploration of distinct approaches to topological quantum matter by leveraging quantum technologies and quantum-inspired classical algorithms. Our results not only advance our understanding of quantum systems but also pave the way for the realization and discovery of novel physics extending to quantum information processing, materials science, and beyond.
In recent decades, the rapid development of quantum technologies has led to a new era of programmable platforms, enabling the realizations of quantum simulation and quantum computation. This dissertation is motivated by recent experimental progress on controlling individual quantum degrees of freedom in systems such as trapped ions and Rydberg atom arrays. By tailoring the interactions in these quantum systems, we study analog quantum simulations of various physical phenomena, including non-equilibrium quantum dynamics and nontrivial topological physics. In the first part of the dissertation, we study slow quantum many-body dynamics in trapped-ion systems and Rydberg atom arrays. We first show that either the long-range interactions or an additional symmetry-breaking field can give rise to a confining potential. Such a potential can couple domain wall quasiparticles into mesonic or baryonic bound states. These confined quasiparticles strongly suppress the quantum information dynamics and lead to slow thermalization. In thelimit of strict domain-wall confinement, the full Hilbert space is fragmented into exponentially many disconnected subspaces. Furthermore, we demonstrate that thermalization can be halted by quantum engineering a uniformly increasing field in the trapped-ion quantum simulator. The second part of the dissertation focuses on topologically relevant phenomena in quantum simulators. We first study the effect of experimentally relevant disorder in 2D Rydberg atom arrays. We find that there are three distinct localization regimes due to the presence of nontrivial topological bands. We further study the non-equilibrium dynamics of Abelian anyons in a one-dimensional system. We show that the interplay of anyonic statistics and interactions can give rise to spatially asymmetric quantum dynamics. Finally, we use Nielsen’s geometric approach to quantify circuit complexity in topological models. We find that the circuit complexities of ground states and non-equilibrium steady states both exhibit nonanalytical behavior at topological transition points.
The recent discovery of topological insulators has led to a tremendous interest in the exploration of topological phases of matter which do not fit into Landau's symmetry breaking paradigm. Numerous exotic topological materials are theoretically predicted. Some of them have been experimentally reported, but many remain not. In this thesis, we explore topological phases of matter from three aspects: their classification, realization and application. We first review some basic classification theories, which provide us a "big picture" and lay the foundation for the rest of the thesis. We then move on to propose a systematic method based on quaternion algebra to construct toy tight-binding Hamiltonians for all the exotic phases in a recently developed periodic table for topological insulators and superconductors. We also introduce two peculiar families of topological phases that are beyond the table---the Hopf and four-dimensional topological insulators without time reversal symmetry. Prototypical Hamiltonians are constructed and their topological properties, such as robust edge states, are numerically studied. Motivated by rapid experimental progress in engineering spin-orbit coupling and artificial gauge field for cold atoms, we continue the thesis by proposing a feasible experimental scheme to realize a three-dimensional chiral topological insulator with cold fermionic atoms in an optical lattice. To unambiguously probe topological phases, we also bring forth systematic and generic methods to measure the characteristic topological invariants, for both free and strongly interacting systems. Moreover, we demonstrate that a kaleidoscope of knot and link structures is encoded in the spin texture of Hopf insulators and show how to observe different knots and links in cold atoms via time-of-flight images. The last part of the thesis is about the application of topological materials. After a demonstration of how to create, braid and detect Majorana fermions with cold atoms, we put forward a proposal to construct a self-test quantum random number generator by using Majorana fermions. Majorana random number generators are able to generate certifiable true random numbers with unconditional security. They offer a new perspective to the utilization of topological materials and may have vital applications in cryptography and related areas.
The ability to manipulate individual quantum systems in a precise way has led to a new era of quantum technologies, including quantum computation and quantum simulation. In this thesis we present several new implementations of these quantum technologies. We first aim at addressing current experimental challenges for quantum dot based quantum computing. We propose a systematic way to study the dynamics of nuclear spin, which is responsible for the short electron spin coherence time. Our calculation is based on diffusion model and is consistent with experiments. We also invent a novel protocol to realize high-fidelity ultrafast universal quantum gate in recently-developed quantum dot molecule system. Experimental realization of our protocol requires only a simple time engineering of optical pulses. We then propose a new quantum state transfer scheme for Nitrogen-Vacancy center based quantum computer, which is applicable at room temperature. Our method accomplishes high fidelity robust quantum state transfer through uncontrolled thermal nitrogen spin chain between two remote NV registers. Our next study helps building a hybrid quantum computer by entangling disparate systems using photonic links. The photons emitted from two types of system need to be matched in both frequency and pulse shape. We propose a simple method to match the emitted pulse shape from two qubit systems with different transition linewidths. We then focus on quantum simulation with trapped ions. We show the possibility of observing a novel type of temperature driven structural phase transition in trapped ion chain, which originates from anharmonic interaction between different vibrational modes. Afterward, an experimental protocol to simulate a conceptually new state of matter, called time crystal, is proposed based on ions trapped in a ring trap. Finally, we propose two new applications based on the recently developed trapped ion quantum simulator of spin models: (1) simulation of Haldane-Shastry model, which opens the way of experimental study to a remarkable theoretical model involving spin liquid ground state and fractional excitations (2) observation of prethermalization and dynamical phase transition, which are poorly understood non-equilibrium phenomena in closed quantum many-body system.
This is an introductory review of the physics of topological quantum matter with cold atoms. Topological quantum phases, originally discovered and investigated in condensed matter physics, have recently been explored in a range of different systems, which produced both fascinating physics findings and exciting opportunities for applications. Among the physical systems that have been considered to realize and probe these intriguing phases, ultracold atoms become promising platforms due to their high flexibility and controllability. Quantum simulation of topological phases with cold atomic gases is a rapidly evolving field, and recent theoretical and experimental developments reveal that some toy models originally proposed in condensed matter physics have been realized with this artificial quantum system. The purpose of this article is to introduce these developments. The article begins with a tutorial review of topological invariants and the methods to control parameters in the Hamiltonians of neutral atoms. Next, topological quantum phases in optical lattices are introduced in some detail, especially several celebrated models, such as the Su–Schrieffer–Heeger model, the Hofstadter–Harper model, the Haldane model and the Kane–Mele model. The theoretical proposals and experimental implementations of these models are discussed. Notably, many of these models cannot be directly realized in conventional solid-state experiments. The newly developed methods for probing the intrinsic properties of the topological phases in cold-atom systems are also reviewed. Finally, some topological phases with cold atoms in the continuum and in the presence of interactions are discussed, and an outlook on future work is given.
This thesis is a contribution to the research field of quantum simulations of topological phases of matter, i.e. quantum states characterized by a non-local order parameter.The aim of the research on quantum simulations is to overcome the difficulties encountered in the study of quantum mechanical models with conventional numerical methods.The proposed way to accomplish this task is the direct exploitation of controllable quantum systems whose properties, for instance low-energy physics, mimic those of the model to be studied.We study several problems connected to the objective of creating a quantum simulator of topological phases of matter.In particular, we discuss how to engineer the Hamiltonian of topological models, how to measure quantities which unravel topological orders, and how to quantify the reliability of topological quantum memories.The first direction of investigation considered in this thesis regards the development of new techniques to simulate topological states of matter with cold atoms in optical lattices.We propose a scheme based on an unconventional use of a bi-chromatic optical lattice to engineer the Hamiltonians of several topological insulators and of the related non-interacting relativistic models.We also focus on the Pfaffian wavefunction, a topological state supporting non-Abelian quasi-excitations arising in the presence of three-body interactions.We present a proposal for the realization with optical lattices of systems characterized by such interactions and discuss the possibility of using such a scheme to create a topological state, such as the mentioned wavefunction.The second line of research considered in this thesis concerns the detection of a string operator in optical lattices.We report on an experiment, to which we provided theoretical support, aimed at the measurement of the non-local order parameter of a one-dimensional Mott insulator; we support the discussion with the relevant theoretical background.Although the Mott insulator is not a topological phase, the measurement of string observables is crucial to reveal the non-local order of several topological states, and the developed experimental protocol is expected to be relevant in future experiments.The third contribution of this thesis is related to the use of topological phases to build a quantum computer, which is a specific example of a quantum simulator.We focus on the important problem of understanding whether it is possible to reliably store quantum information in topological states.We consider one specific class of topological quantum memories, namely those based on a fermionic topological superconductor with zero-energy Majorana modes.At the end of the storage period, we take into account the possibility of applying an operation aimed at the recovery of the information initially encoded into the system and then diminished by the time evolution.In particular, we vii